AN EFFICIENT NUMERICAL METHOD FOR CAVITATION IN NONLINEAR ELASTICITY
XIANMIN XU† AND DUVAN HENAO‡
†INSTITUTE OF COMPUTATIONAL MATHEMATICS AND SCIENTIFIC/ENGINEERING COMPUTING, CHINESE ACADEMY OF SCIENCES, BEIJING 100190, CHINA.
‡LABORATOIRE JACQUES-LOUIS LIONS, UMR 7598, CNRS - UNIV. PIERRE ET MARIE CURIE, BOˆITE COURRIER 187, 75252 PARIS CEDEX 05, FRANCE.
Abstract. This paper is concerned with the numerical computation of cav- itation in nonlinear elasticity. The Crouzeix-Raviart nonconforming finite element method is shown to prevent the degeneration of the mesh provoked by the conventional finite element approximation of this problem. Upon the addition of a suitable stabilizing term to the elastic energy, the method is used to solve cavitation problems in both radially symmetric and non-radially symmetric settings. While the radially symmetric examples serve to illustrate the efficiency of the method, and for validation purposes, the experiments with non-centred and multiple cavities (carried out for the first time) yield novel observations of situations potentially leading to void coalescence.
1. Introduction
Motivated by the study of the internal rupture of rubber cylinders of Gent and Lindley [13], a number of researchers in nonlinear elasticity have considered the problem of modelling the formation and rapid expansion of voids in solids subjected to tension, phenomenon referred to in the literature as cavitation. In the mathematical theory that has developed, hole-creating deformations are obtained as singular minimizers of the stored energy of the material, whenever sufficiently large tensile loads are applied to the body (see, e.g., the seminal work by Ball [4], the review paper by Horgan and Polignone [20], or the more recent works by M¨uller and Spector [28], Sivaloganathan and Spector [36], Conti and De Lellis [10], and Henao and Mora-Corral [16, 17, 18]).
2000 Mathematics Subject Classification. 65N30,74B20,74S05.
Key words and phrases. Cavitation, nonconforming FEMs, gradient flow.
†Corresponding author.
Many difficulties arise in the numerical computation of cavitation, espe- cially because the corresponding minimization problems exhibit the Lavrentiev phenomenon [22], making the traditional finite element methods unable to de- tect the singular minimizers of the stored energy. Although several methods have been proposed to overcome the Lavrentiev phenomenon [2, 6, 21, 23, 32], they have not been used for the simulation of cavitation (except for the radially symmetric case, see Knowles [21]). This is due not only to the nonconvexity and high nonlinearity of the energy functional, but also to the large deformations involved (near the cavity surfaces) and the presence of point singularities.
In order to overcome the mentioned numerical difficulties, we do not con- sider the full minimization problem, in which all forms of cavitation are allowed to compete in the minimization of the energy, but the simpler situation of the model of Sivaloganathan and Spector (see [36, 37, 38, 40, 41], among others), where a bound is imposed on the number of cavities, and the possible locations of the singularities in the reference configuration are prescribed. By the ap- proximation result of Sivaloganathan, Spector and Tilakraj [39] (see also [15]), we obtain the singular minimizers for this problem, in the limit as ǫ → 0, by minimizing the elastic energy, in a class of regular deformations, on a domain with holes of radius ǫ centred at the prescribed cavity points.
For the case of a single cavity at the centre of a ball, Negr´on-Marrero and Betancourt [29] were able to carry out computations for the model described above, allowing for non-radially symmetric deformations, by using a spectral- collocation method. For more complex cavitation configurations, however, the efficiency of the traditional numerical methods is affected by the inability to accurately resolve the singularities of the deformation, and by the propensity of these methods to induce the degeneration of the mesh. The purpose of this pa- per is to show that the singular minimizers arising in the cavitation models can be numerically obtained by using instead the Crouzeix-Raviart nonconforming finite element method [11], as previously suggested by Ball [7, p. 6] (for further computations of cavitation in domains with multiple holes, based on a quadratic iso-parametric finite element method, see also the recently completed works by Lian and Li [24, 25]).
The structure of this paper is as follows. In Section 2, we briefly introduce the original and regularized energy models. In Section 3, we discuss the finite
element approximation of the regularized model. Both the mesh degeneracy of the conforming finite element method, and the lack of this degeneracy in the nonconforming method, are carefully analyzed by means of a scaling argument.
Following this analysis, we show that the nonconforming method is prone to its own numerical instability, but that this can be controlled by adding a suitable stabilizing term to the elastic energy. In Section 4, we introduce the corre- sponding numerical algorithm for the stabilized problem, where minimizers are sought by means of a suitably chosen gradient flow. In Section 5, we present some radially-symmetric numerical experiments, in order to illustrate the effi- ciency and the correctness of our method, and present some new observations of cavitation obtained, for the first time, in off-centre configurations and in domains containing multiple cavities.
2. The energy model for cavitation in nonlinear elasticity 2.1. General notation. We suppose that a nonlinearly elastic body occupies the domain Ω ⊂ Rn, n = 2,3, in its reference configuration. Deformations of the body are represented by maps u : Ω 7→ Rn, where u(x) denotes, for each x∈Ω, the position ofx in the deformed configuration. The map u is assumed to be weakly differentiable, and its gradient is denoted by ∇u(x).
Vector-valued and matrix-valued quantities are written in boldface. The identity matrix is denoted by 1. Given a square matrixF∈Rn×n, its transpose is denoted by FT, and its determinant by detF. The cofactor matrix of F, denoted cofF, is the matrix satisfying FT cofF = (detF)1. The dot product of two vectors a,b ∈Rn is denoted by a·b, and the same notation is used for the inner product of matrices.
2.2. The model without pre-existing microcavities. The elastic stored energy of the body is given by
I(u) = Z
Ω
W(∇u)dx, (2.1)
whereW is the energy density function. Model examples include the compress- ible neo-Hookean materials, given by
W(F) = µ
2(|F|2−2 ln detF−3) (2.2)
(µ >0 being the shear modulus), or energy densities of the form W(F) = µ
p|F|p+f(detF), (2.3) where p > 1 and f : (0,+∞) → R is a suitable convex function. The role of f, in general, is to prevent the compression of a subpart of the body to zero volume, or to penalize deviations from incompressibility. For this type of energy densities, certain conditions are required for cavitation to take place [4, 28]. We suppose that n−1< p < n, and that
s→0lim+f(s) = lim
s→+∞
f(s)
s = +∞. (2.4)
In order to guarantee the existence of minimizers of the energy, deformations are assumed to belong to the Sobolev space (W1,p(Ω))n, and to satisfy certain invertibility conditions, such as condition (INV) by M¨uller and Spector [28], or condition (INV′) in [15]. For simplicity, we do not give the details of these conditions, but refer to the cited articles.
In this paper we consider the model of Sivaloganathan and Spector [36], where cavitation is allowed only at a finite number M ∈ N of prescribed loca- tions x1, . . . ,xM. This is achieved by imposing that
Det∇u= (det∇u)Ln+
M
X
i=1
αiδxi, (2.5)
where Ln is the n-dimensional Lebesgue measure, δxi is the Dirac measure supported on xi, and the αi ≥ 0 are the volumes of the cavities. Here Det∇u denotes the distributional determinant of u (see [3, 27]), which is defined as
Det∇u(φ) :=−1 n
Z
Ω
u·(cof∇u)∇φdx, φ ∈C0∞(Ω). (2.6) Upon specifying appropriate boundary conditions, such as u=g on∂Ω for suitable g:∂Ω→Rn, the model can be formulated as
min
u∈V I(u), (2.7)
where
V ={u∈(W1,p(Ω))n|det∇u>0,u satisfies (INV),u|∂Ω =g, (2.8) Det∇u = (det∇u)Ln+
M
X
i=1
αiδxi}.
2.3. The energy model with pre-existing microcavities. The minimiza- tion problem (2.7) exhibits the Lavrientiev phenomenon, whereby
min
u∈V∩(W1,∞)nI(u)>min
u∈V I(u),
despite the fact that V ∩(W1,∞)n is dense inV. Because of this, the traditional finite element method fails to give the minimizers and the minimum value of the energy for this problem. In order to overcome this difficulty, we consider the regularized model of Sivaloganathan, Spector and Tilakraj [39], where small holes of radius ǫ centred at the cavity points are removed from the domain.
Letting Ωǫ := Ω\PM
i=1B(xi, ǫ), this can be written as min
u∈Vǫ
Iǫ(u), (2.9)
where Iǫ and Vǫ are given by
Iǫ(u) = Z
Ωǫ
W(∇u)dx, (2.10)
Vǫ ={u ∈(W1,p(Ωǫ))n|det∇u>0,u satisfies (INV),u|∂Ω =g, (2.11) Det∇u= (det∇u)Ln}.
As shown in [39, 15], (2.9) converges to (2.7) in the sense of Γ-convergence.
3. Finite element approximations of the relaxed problem The purpose of this section is to describe some obstacles encountered in a preliminary attempt to carry out computations for (2.9) using the conforming finite element method, and to explain how they were overcome by following instead the Crouzeix-Raviart [11] nonconforming approach.
3.1. The finite element methods. Let Thǫ be the triangulation of Ωǫ. We suppose that all the triangles satisfy the maximum angle principle [1]. We can approximate Ωǫ with a polygonal domain, since this introduces only a higher order error. We suppose, thus, that Ωǫ is a polygonal domain such that Ωǫ =
∪K∈ThǫK. Define the linear conforming finite element space Vhc Vhc =
v:v|K ∈(P1(K))n,det∇v|K >0,∀K ∈ Thǫ,
v∈(C(Ωǫ))n,v(ai) =g(ai),∀ai ∈∂Ω}, (3.1) and the Crouzeix-Raviart nonconforming finite element space
Vhnc =
v:v|K ∈(P1(K))n,det∇v|K >0,∀K ∈ Thǫ,
v is continuous on M,v(aij) =g(aij),∀aij ∈ M ∩∂Ω},(3.2) where M denotes the set of mid-points of the edges inThǫ.
The conforming and nonconforming finite element approximations of (2.9) are defined, respectively, as
min
uh∈VhcIhǫ(uh), (3.3) and
min
uh∈VhncIhǫ(uh). (3.4) Here Ihǫ(uh) is the discrete energy functional
Ihǫ(uh) = X
K∈Thǫ
|K| W(∇uh|K). (3.5)
3.2. Mesh degeneracy of the conforming finite element method. By means of a scaling argument, we show how the efficiency of the conforming method is affected by its propensity to make the discrete energy unbounded, and to result in a certain form of mesh degeneracy. For simplicity, consider a domain with a single cavity at the center, Ωǫ = Ω\B(0, ǫ), and a triangulationThǫ of the domain, where the inner boundary is approximated byN line segments. Forr >
0 and i∈N, denote the point with polar coordinates (r,πiN) byAr,i. We assume that ∂B(0, ǫ) is approximated by the segments Aǫ,i−2Aǫ,i, i = 2,4, . . . ,2N. Furthermore, suppose that the first layer next to the cavity surface is composed of the following two sets of triangles (see Figure 3.1):
• those with vertices at Aǫ,i−1, Aǫ+δ,i and Aǫ,i+1, for i= 1,3, . . . ,2N −1;
and
• those with vertices at Aǫ+δ,i−1, Aǫ,i and Aǫ+δ,i+1, for i= 2,4, . . . ,2N. In both cases, the triangles are denoted Ki, i= 1,2, . . . ,2N; the height of the triangles is proportional to the parameter δ >0.
K1 K3
K2 K4
K5 K7
K6
Aε,0 Aε,2 Aε,8
Aε,6
Aε,4
Aε+δ,1 Aε+δ,7
Aε+δ,5
Aε+δ,3
Figure 3.1. VerticesAr,i in a triangulation of a domain with microcavities.
When using the conforming element method, great care is needed in choos- ing N sufficiently large so that the polygon formed by the nodes on the cavity surface accurately resolves the curvature of that surface not only in the ref- erence, but also in the deformed configuration. As schematically described in Figures 3.2 and 3.3, some of the elements in the first layer may reverse their orientation during the expansion of a microcavity. Figure 3.2 shows the polygon Aǫ,0Aǫ,2· · ·Aǫ,2N, the circle ∂B(0, ǫ+δ), and the base of an element Ki, with i even. The figure in the middle corresponds to a uniform dilation, while the figure on the right shows a further compression in the radial direction. Figure 3.3 represents the same situation, but focuses on the elements themselves. The reason for this degeneracy is that the distance from a curved surface to the mid-point of a line segment with its ends on the same surface will be consider- ably large compared to the height of the deformed elements (unless the distance between the two end-points is small enough). The remedy is to use very fine meshes, which affects the efficiency of the method. In the calculations that follow, we study the P1-interpolation of a radially symmetric incompressible deformation, and conclude that δ andN−1 should both go to zero, when ǫ→0, at least as ǫp−1.
Figure 3.2. Orientation reversal of elements under the conform- ing approximation.
Figure 3.3. Initial and final state of elements at the cavity sur- face, corresponding to Figure 3.2.
Let u : Ωǫ → R2 be a radially symmetric and incompressible deformation opening a cavity of radius c >0. Writing u(x) = u(|x|)|xx|, where u is of class C1, the incompressibility condition reads u′(s)u(s) = s. Thus,
u(x) =p
|x|2+c2 x
|x|. (3.6)
Suppose now that u is approximated by its interpolation in Vhc, given by
uh|K =
3
X
i=1
u(ai,K)φi,K, ∀K ∈ Thǫ,
where ai,K, i = 1,2,3, are coordinates of the nodes of K, and the functions φi,K ∈P1, which are such thatφi,K(aj,K) = δij, are the order-one finite element base functions on the triangle K.
Assume the stored-energy function W is of the form (2.3). By considering the elements in the first layer, we observe that
Ihǫ(uh) ≥
N
X
l=1
|K2l−1|W(∇uh|K2l−1)≥ µ p
N
X
l=1
|K2l−1|
(∇uh|K2l−1)
p
≥ µ p
N
X
l=1
|K2l−1|
∇uh a2l−a2l−2
|a2l−a2l−2|
p = µ p
N
X
l=1
|K2l−1|
uh(a2l)−uh(a2l−2)
|a2l−a2l−2|
p.
Here ai denotes the position vector of Aǫ,i, for i = 0,2, . . . ,2N. A straightfor- ward computation shows that
|K2l−1|=ǫsin π
N(δ+ 2ǫsin2 π
2N), l = 1,2, . . . , N;
uh(a2l)−uh(a2l−2)
|a2l−a2l−2|
= u(ǫ) ǫ ≥ c
ǫ, l = 1,2, . . . , N.
For N ≥ 2 we have that sinNπ ≥ 2Nπ , hence Ihǫ(uh) ≥ πµ2pcpδǫ1−p. We conclude that, in order for the energy in the first layer to remain bounded, the choice of δ as a function of ǫ must be such that δ=O(ǫp−1).
Consider now the potential degeneration of the elements on the cavity sur- face. The image of Ki, i = 2,4, . . . ,2N, under the deformation uh, is the triangle with vertices at
(u(ǫ+δ),(i−1)Nπ), (u(ǫ), iNπ), and (u(ǫ+δ),(i+ 1)Nπ).
Although u(ǫ+δ) > u(ǫ), it is possible for the deformed triangle uh(Ki) to become increasingly flat, or even to reverse its orientation (as in Figure 3.3), if N(ǫ)6→ ∞. Indeed, suppose N2 <2c2/δ(2ǫ+δ), then
δ(2ǫ+δ) 2c < π2
8 N−2p
(ǫ+δ)2+c2
⇒ p
(ǫ+δ)2+c2−√
ǫ2+c2 <2p
(ǫ+δ)2+c2sin2 π 2N
⇒ u(ǫ+δ) cos π
N < u(ǫ).
Hence N must be larger than √ 2c/p
δ(2ǫ+δ). Since δ = O(ǫp−1) and p < 2, we obtain N−1 =O(ǫp−1). Hence the conforming method requires the mesh to become extremely fine, both in terms of the height of the triangles and of the number of elements at the cavity surface.
3.3. Scaling analysis for the Crouzeix-Raviart finite element method.
Denote by u the radial incompressible deformation (3.6), and let ˜uh be its in- terpolation in the nonconforming finite element space Vhnc. If, given an element K ∈ Thǫ, we callpi,K,i= 1,2,3 the mid-points of its sides, and if we denote the order-one nonconforming finite element base functions on K byϕi,K, i= 1,2,3 (so that ϕi,K(pj,K) = δij), then ˜uh is given by
˜ uh|K =
3
X
i=1
u(pi,K)ϕi,K ∀K ∈ Thǫ. (3.7) Suppose that the triangulation in the layer next to the cavity surface is as in the previous subsection. Denote the mid-points of the sides of the elements K1, . . . , K2N by p1, . . . ,p4N+1, with p1 = p4N+1, as depicted in Figure 3.4.
More precisely, if we let r0 =
s ǫ
2cos π
N +ǫ+δ 2
2
+ǫ 2sin π
N 2
, θ0 = arctan ǫsinNπ ǫcos Nπ +δ+ǫ,
(3.8) then the mid-points p2i−1, p2i,p2i+1 of the sides of Ki have polar coordinates
(r0,iπN −θ0), (ǫcosNπ,iπN), (r0,iπN +θ0) i= 1,3, . . . ,2N −1, (r0,(i−1)πN +θ0), ((ǫ+δ) cosNπ,iπN), (r0,(i+1)πN −θ0) i= 2,4, . . . ,2N.
K1 K3
K2 K4
K5 K7
K6
P6 P5 P4
P3 P2
P1 P9 P8
P7 P13
P12
P11 P10 P14
P15
Figure 3.4. Mid-points of the edges in a triangulation of a do- main with microcavities.
The main reasons for the choice of the nonconforming method are that the triangles Ki do not degenerate or reverse their orientation under ˜uh, and that
the energy of ˜uh remains close to that of u (as opposed to the energy of uh), even if N and δ are kept fixed as ǫ → 0. Regarding the mesh degeneracy, the most important feature of the method is that by imposing continuity only at the mid-points, and not on the edges in their entirety, it gives each element on the cavity surface the liberty to retain its true orientation, the one dictated by the expansion of the cavity and by the minimization problem. This, which is illustrated in Figures 3.5 and 3.61, contrasts with the situation in the conforming method, where the orientation of the triangles Ki, withi odd, was forced upon those with i even2. Regarding the contribution to the energy of the elements next to the cavity, the main virtue of the method is that the tangential stretch, which constitutes the largest component of the deformation gradient, is given by the stretch of the segment joining the mid-points of the elements. Because of this, it is always of order c/(ǫ+δ2), wherecis the cavity radius, and so remains bounded asǫ→0, even for constant δ andN. These assertions will be justified and clarified in the subsequent analysis.
Figure 3.5. Effect of the conforming and nonconforming finite element methods on the orientation of elements on the cavity surface.
In order to analize the asymptotic behaviour of the relevant quantities for small values ofǫ, and since N and δwill be treated as constants, it is important to precise in what sense certain terms (such as δ4 orδ2N−2) can be considered of higher order, or comparatively smaller, with respect to the main terms in the asymptotic expansions. Before proceeding further, therefore, we specify that expressions of the form
f(ǫ, δ, N) =g(ǫ, δ, N) +O(h(ǫ, δ, N))
1The elements in Figure 3.6 ressemble lines in the deformed configuration due to the radial compression (of the order of 10−2). It is observed numerically that the discrete solution is orientation preserving (the determinant is approximately 1.466 in all of the elements, see Section 3.5). Also, we note that the angle made by the base of each element with respect to the curved cavity surface in the deformed configuration is close to the angle made in the reference configuration.
2In the sense that the direction, in the deformed configuration, of the segmentsAǫ+δ,i−1Aǫ,i andAǫ+δ,i+1Aǫ,i, fori even, is determined by the image ofuh(Ki−1) anduh(Ki+1).
0.01 0.015 0.02 0.025 0.03 0.035
−0.02
−0.015
−0.01
−0.005 0 0.005 0.01 0.015 0.02 0.025
0.9 1 1.1
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4
Figure 3.6. Elements in the computations of Section 5, initial and final states (ǫ= 0.01, δ≈0.025, N = 15, λ= 1.8).
(such asu(r0) = c+2c1(ǫ+δ/2)2+O (ǫ+δ)4
, orr0 =ǫ+δ/2 +O ǫN−2 ) will be given the following meaning: for some δ0 > 0 and N0 ∈ N there exists, for every δ < δ0 and every N > N0, constantsǫ0(δ, N)>0 and C =C(δ0, N0)>0 such that
ǫ < ǫ0(δ, N) ⇒ |f(ǫ, δ, N)−g(ǫ, δ, N)|< Ch(ǫ, δ, N).
We study first the behaviour of ∇u˜(N)h |Ki for i even. Writting e1 := (cosiπ
N,siniπ
N), e2 := (−siniπ
N,cosiπ
N), p˜ = p2i−1+p2i+1
2 (3.9)
we have that
(∇u˜h|Ki)e1 = u(p2i)−u(˜p)
|p2i −p˜| = u (ǫ+δ) cosNπ
−u(r0) cos(Nπ −θ0)
(ǫ+δ) cosNπ −r0cos(Nπ −θ0) e1 (3.10)
(∇u˜h|Ki)e2 = u(p2i+1)−u(p2i−1)
|p2i+1−p2i+1| = 2u(r0) sin(Nπ −θ0)
2r0sin(Nπ −θ0) e2 = u(r0) r0
e2. (3.11) Letai,ai−1 denote the position vectors ofAǫ,i, Aǫ+δ,i−1, respectively. From (3.8) we deduce that
r0cosθ0 = (ǫ+δ/2)−ǫsin2 π
2N, sinθ0 =ǫsin π
N/2r0, (3.12) (ǫ+δ)/2< r0 =|ai+ai−1|/2<(|ai|+|ai−1|)/2 =ǫ+δ/2. (3.13) Using that sin2Nπ < 2Nπ we obtain
r0 =r0cosθ0+ 2r0sin2 θ0
2 = (ǫ+δ/2) +O(ǫN−2). (3.14) This, combined with |√
1 +t−(1 +t/2−t2/8)| <|t3/2|, for t∈ (−1,∞), and with |t−sint|<|t3/6|, for all t∈R, yields
π
N −θ0 = π
N 1 +O ǫ/(ǫ+δ)
, 1
r0
= 2δ−1(1 +O ǫδ−1
) (3.15)
u(r0) = c+ 1
2c(ǫ+δ/2)2 − 1
8c3(ǫ+δ/2)4+O((ǫ+δ)6). (3.16) From (3.15), (3.16), and (3.11) we find
u(r0)
r0 = 2cδ−1(1 + 1
8c2δ2+O(δ4)). (3.17) Using once again the estimate for √
1 +t, we have that (ǫ+δ) cos π
N = (ǫ+δ)(1−2 sin2 π
2N) = (ǫ+δ)(1 +O(N−2)), (3.18) u (ǫ+δ) cos π
N
=c+ 1
2cδ2− 1
8c3δ4+O(δ6) +O(δ2N−2). (3.19) As a consequence of (3.15), we have that cos(Nπ −θ0) = 1 +O(N−2). Hence
u(r0) cos(π
N −θ0) =c 1 + 1
8c2δ2− 1
128c4δ4+O(δ6) +O(N−2)
. (3.20) In order to find (∇u˜h|Ki)e1 it only remains to compute r0cos(Nπ −θ0), which, by virtue of (3.14), is given by
r0cos(π
N −θ0) = (ǫ+δ/2)(1 +O(N−2)). (3.21)
Combining (3.10), (3.19), (3.20), (3.18), and (3.21) we arrive at
(∇u˜h|Ki)e1
=c−1δ 3
4 − 15
64c2δ2+O(δ4) +O(δ−2N−2)
. (3.22)
For the case of Ki with i odd, defining e1 and e2 as in (3.9) we have (∇u˜h|Ki)e1 = u(r0) cosθ0−u(ǫcosNπ)
r0cosθ0−ǫcosNπ e1, (∇u˜h|Ki)e2 = u(r0) r0
e2.
Proceeding as before, we obtain cosθ0 = 1 +O ǫ2N−2/(ǫ+δ)2 , u(r0) cosθ0−u(ǫcos π
N) = δ2 4c
1− δ2
32c2 +O(δ4)
1
r0cosθ0−ǫcosNπ = 2δ−1 1 +O ǫN−2δ−1 (∇u˜h|Ki)e1 = δ
4c
1− 1
16c2δ2+O(δ4)
(3.23) In both cases (see (3.22), (3.23)), (∇u˜h|Ki)e1 = O(δ). From (3.14) and (3.16) we obtain
u(r0)
r0 = 2c
2ǫ+δ 1 +O (ǫ+δ)2 .
Hence, for i= 1,2, . . . ,2N,
|∇u˜h|Ki|p =
u(r0) r0
p
1 +
(∇u˜h|Ki)e1
u(r0)/r0
2!p/2
=
2c 2ǫ+δ
p
1 +O (ǫ+δ)2
1 +O δ2(ǫ+δ)
= 2pcp(2ǫ+δ)−p 1 +O (ǫ+δ)2
. (3.24)
This estimate allows us to compare E1 =
Z
ǫ<|x|<ǫ+δ|∇u|pdx=π
Z (ǫ+δ)2
ǫ2
2 + c4 t(t+c2)
p2 dt
against E1h =
2N
X
i=1
|Ki| |∇u˜h|Ki|p =N|K1| |∇u˜h|K1|p+N|K2| |∇u˜h|K2|p.
Using that
2 + c4 t(t+c2)
p2
=cpt−p2 +O(t1−p2), we obtain E1 = 2πcp
2−p(ǫ+δ)2−p
1− ǫ/(ǫ+δ)2−p
+O δ(ǫ+δ) .
As for E1h, notice that
|K1|=ǫδπ
N(1 +O(N−2)), |K2|=δ(ǫ+δ)π
N(1 +O(N−2)), hence
E1h = 2pπcpδ(2ǫ+δ)1−p 1 +O(N−2) +O((ǫ+δ)2) .
We conclude that E1 and E1h are of the same order of magnitude (namely, of order δ2−p), in contrast to the situation in the conforming method, where E1h → ∞ asǫ→0.
3.4. Instability of the nonconforming finite element method. Some care is needed when using the Crouzeix-Raviart method, due to its propensity to ob- tain pathological solutions to the equilibrium equations as the one depicted in Figure 3.7. In this figure the elements near∂Ω stretch in one direction (because of the boundary condition), but compress in the other (so as to preserve the volume of the elements), creating discontinuities throughout the mesh in its entirety3. These solutions are obtained because of the flexibility of the method, where the elements can choose their own orientation independently of the orien- tation of their neighbouring elements. As explained in the previous subsections, such behaviour is desirable on the cavity surface, but it is inadmissible in the interior and on the outer boundary.
Similar instabilities have been observed for problems in linear elasticity and in fluid dynamics (see, e.g., [12, 9]), and are related to the lack of a dis- crete Korn’s inequality in the nonconforming setting. For these problems the Crouzeix-Raviart method is stabilized by penalizing the jump of the discrete
3The instability continues to be observed even for very fine meshes.
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
(a) Reference configuation
−1.8 −1.4 −1 −0.6 −0.2 0.2 0.6 1 1.4 1.8
−1.8
−1.4
−1
−0.6
−0.2 0.2 0.6 1 1.4 1.8
(b) Deformed configuration
Figure 3.7. Instability of the Crouzeix-Raviart finite element method; ǫ= 0.01, stored-energy function (5.1).
function on the sides of all inner elements. However, numerical experiments reveal this to be insufficient in the cavitation problem. Figure 3.8 shows a numerical solution to (3.4) using the L2 penalty term
C h
X
E∈E
Z
E|[uh]|2ds, (3.25)
where C is a constant, h and E are the size parameter and the set of sides of the triangulation, and [·] denotes the jump of a given quantity across the corresponding side. Even though the solution is correct near the boundary, the pathological behaviour persists in the interior of the body. This can only be solved by increasing the value of C considerably as the mesh becomes finer.
For the nonlinear problem under consideration, we stabilize the Crouzeix- Raviart approximation by adding
C h
X
E∈E
Z
E|[∇uh·s]|2ds (3.26)
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
−1.8 −1.4 −1 −0.6 −0.2 0.2 0.6 1 1.4 1.8
−1.8
−1.4
−1
−0.6
−0.2 0.2 0.6 1 1.4 1.8
Figure 3.8. Instability of the Crouzeix-Raviart method using theL2penalty term (3.25);ǫ= 0.01, stored-energy function (5.1).
to the discrete energy, penalizing the orientation mismatch between neighbour- ing elements across the sides. Here C, h, E, and [·] are as in (3.25)4, and s denotes one of the two unit vectors in the direction of E (its orientation be- ing suitably specified). With this penalty term the behaviour in Figures 3.7 and 3.8 becomes expensive because [∇uh ·s] remains essentially of the same magnitude, independently of the smallness of the elements. Indeed, |[∇uh·s]| depends only on the difference in direction, in the deformed configuration, be- tween corresponding sides of adjacent elements (in contrast, e.g., to|[uh]|, which is proportional to the length of the edges). The price to pay is thus of the order of h−1P
E∈Eℓ(E)∼h−1nEh ∼h−2, where we have denoted byℓ(E) the length of E and by nE the total number of elements.
In our numerical experiments, the penalty term is observed to provide the stability necessary for the method to find the true solutions of the minimiza- tion problem, rather than the unexpected local minimizers described previously.
Nevertheless, the analysis of the penalization scheme remains an important open problem, both for discontinuous Galerkin methods and for nonconforming finite element methods. In the case of cavitation, we may argue that |[∇uh·s]| is of
4The actual values of C and h in our computations being 0.05 and n−1/2
E , respectively, where nE denotes the total number of elements.
the order of δ−1sinNπ near the cavity surface, and, hence, that the penalty term for the true solution to the problem (using nonconforming elements) is of the order of
h−1(δ−1N−1)2N(δ+N−1) = N−1
hδ (1 +hN−1 hδ )
(the last two factors on the left-hand side corresponding to the number of el- ements in the first layer, and to their perimeters, respectively). This suggests that the price to pay, in the actual solution, for the orientation mismatch of the elements on the cavity surface (depicted in Figure 3.5 and discussed in Section 3.3) is sufficiently small, at least if N−1 ≪ hδ. Numerically, we observe that the penalty term is indeed small, even in situations where the condition on N, h, andδ is not satisfied, such as the one in Figure 3.6 in which δ ≈ 0.025 and N = 15.
3.5. Jacobian determinant at the cavity surface. From the work of Ball [5] (see also [8, Sect. 2.4], [36, Thm. 5.1]), it follows that minimizers of (2.10) satisfy the Eulerian form of the equilibrium equations:
Z
Ω
∇W ∇u(x)
∇u(x)T
· ∇ϕ u(x)
dx= 0
for all ϕ∈C1(Rn,Rn) such thatϕ and ∇ϕ are uniformly bounded and satisfy ϕ◦u|∂Ω = 0 in the sense of traces. Ifuis sufficiently regular in Ωǫ, so thatu(Ωǫ) is open and the cavities have sufficiently regular boundaries, the equilibrum equations imply that T(y)ν(y) = 0 at every y on the surface of any of the cavities, where Tstands for the Cauchy-stress tensor
T(y) = ∇W(∇u(x))∇u(x)T(det∇u(x))−1, y=u(x),
and ν(y) represents the unit normal to the cavity. Taking the dot product with ν, and applying this condition to the energy function (2.3), we obtain
f′(det∇u(x)) = −µ|∇u(x)|p−2
∇u(x)T
det∇u(x)ν(u(x))
2
det∇u(x)
x ∈ ∂Bǫ(xi), i = 1, . . . , M. From the convexity of f and the fact that f(s) is increasing for large values of s, it follows that det∇u ≤ s∗ at the cavity surface, where s∗ denotes the minimizer of f. Note that det∇u∇Tu = cof∇u−1, and that |(cof∇u−1(y))ν(y)|gives the infinitesimal change in area, due tou−1,
at the cavity surface. If the expansion of the cavities is uniform, in the sense that |(cof∇u−1)ν|is approximately constant on the boundary of the cavity, we obtain that
|f′(det∇u)| ≤Cc ǫ
p−2−2(n−1)
,
where c denotes the radius of the cavity. From this we observe that det∇u converges to s∗, as ǫ→0, at every point on the cavity surface.5
−1
−0.5 0
0.5
1 −1
−0.5 0
0.5 1
1.4 1.6 1.8 2 2.2 2.4 2.6
−0.02
0 0.02 −0.02 0 0.02
1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66
Figure 3.9. Jacobian determinant of the numerical solution us- ing the Crouzeix-Raviart finite element method.
The previous is a formal calculation, but numerically it is indeed observed that det∇u ≈ s∗ on the cavity surface. For the energy function (5.2), where f(s) = 12(s−1)2 +s−1, the numerical solution uh is such that det∇uh|Ki is close to 1.4656 (the minimizer of f) for every i = 1, . . . ,2N (see Figure 3.9).
This contrasts with the interpolation ˜uh of the radial incompressible solution (studied in Section 3.3), where
det∇u˜h|Ki = 3 2 − 9
32c2δ2+O(δ4) +O(δ−2N−2), i= 2,4, . . . ,2N, det∇u˜h|Ki = 1
2 + 1
32c2δ2+O(δ4), i= 1,3, . . . ,2N −1
(the expansions are obtained by multiplying (3.22) and (3.17), considering that {e1,e2} is an orthonormal basis). The reason for this discrepancy is that the
5The argument is based on [4, Sect. 7.5]; see also [30, Eq. 33].
numerical solution will adjust itself, abandonning the pointwise convergence to the interpolation ˜uh at the nodes close to the cavity surface, in order to favour the convergence of the Jacobian determinants. Given the structure of the energy function, and its direct dependence on the determinant, the convergence of this quantity becomes much more relevant for the minimization of the energy.
The validity of the scaling analysis of the previous subsections is not af- fected by the above discrepancy between uh and ˜uh, since the slight difference between uh(pi) and ˜uh(pi), for i = 1, . . . ,4N, does not have a great effect on the estimates for the tangential derivatives, which are the most relevant for the study of the contributions to the energy. What is important (as mentioned after Equation (3.8)), is that the tangential stretch u(r0)/r0 is computed at the mid- points of the elements, hence it is of order 2cδ−1 (cbeing the cavity radius), and not cǫ−1. This makes the method efficient in that it does not require excessively fine meshes.
Since det∇uhis close to 1.4656 in the elements close to the cavity (as proved numerically), these elements do not become flat or reverse their orientation as under the conforming method. The flexibility expected of the nonconforming method for the orientation of the elements close to the surface (illustrated in Figure 3.5), is also corroborated numerically.
We end by noting that the average determinant in the inner layer, (32+12)/2, coincides with the determinant of the radial incompressible deformation. Both the oscillating behaviour of det∇u˜h|Ki and the before-mentioned agreement should be taken into account in a rigorous proof of convergence of the non- conforming method for the problem of cavitation, and potentially for other nonconvex variational problems with singular minimizers.
4. The numerical method and gradient flows
4.1. The gradient flow equations. We look for minimizers of (2.9) by solving its corresponding H01 gradient flow
−∆ut(x, t) = div(∇W(∇u(x, t))), x∈Ωǫ, t >0 u(x, t) = g(x), x∈∂Ω, t >0
(∇ut+∇W(∇u))ν(x) = 0, x∈∂Bǫ(xi), i= 1, . . . , M, t >0
(4.1)
(ν(x) being the unit normal to ∂Bǫ(xi)) in its weak form given by Z
Ωǫ
∇ut·∇v=− Z
Ωǫ
(µ|∇u|p−2∇u+f′(det∇u)cof∇u)·∇vdx ∀v∈V0, (4.2) with V0 := {v ∈ (W1,p(Ωǫ))n|v = 0 on ∂Ω}. Equation (4.1) gives a steepest descent iteration for minimizing the stored energy functional, as seen from
dI(u)
dt =
Z
Ωǫ
(µ|∇u|p−2∇u+f′(det∇u)cof∇u)· ∇utdx
= −
Z
Ωǫ
∇ut· ∇utdx≤0, (4.3)
and is known to provide a faster and more stable numerical scheme than, for example, the L2 gradient flow
ut = div(µ|∇u|p−2∇u+f′(det∇u)cof∇u) (4.4) (see, e.g., [31, Ch. 2 and Sect. 11.6]). In their computations of cavitation, Negr´on-Marrero and Betancourt [29] have used a Richardson extrapolation tech- nique, based on the equation
utt+ηut = div(µ|∇u|p−2∇u+f′(det∇u)cof∇u), η≪1, (4.5) for which the Courant-Friedrichs-Lewy condition (see, e.g., [26]) is not as re- stringent as for (4.4). Denoting the time step by δtand the mesh size byh, the CFL condition for (4.5) and for (4.4) reads δt= O(h) and δt=O(h2), respec- tively. Due to the presence of the Laplacian operator in front of ut, in the case of (4.1) the condition becomes δt=O(1). This has motivated the choice of our method, whose efficiency is comprobated in the numerical experiments.
4.2. The discretization and the numerical algorithm. To discretize the above equations, we introduce the finite element space
Vh,0nc =
v:v|K ∈(P1(K))n, v is continuous on M,
v(aij) = 0 ∀aij ∈ M ∩∂Ω}. (4.6) We discretize (4.2) in space using the nonconforming finite element method.
Taking into account the stabilization term (3.26), we obtain the semi-discrete
equation X
K∈T
Z
K∇duh
dt · ∇vh = − X
K∈T
Z
K
(µ|∇uh|p−2∇uh+f′(det∇uh)cof∇uh)· ∇vhdx
−C h
X
E∈E
Z
E
[∇uh·s][∇vh·s]ds, ∀vh ∈Vh,0nc. (4.7) Let now 0 = t0 < t1 < t2 < . . . < tN = T < ∞, and replace the time derivative at each tn by the forward difference
duh
dt |t=tn ≈ un+1h −unh tn+1−tn
.
For n= 0,1,· · · we obtain X
K∈T
Z
K∇un+1h −unh tn+1−tn · ∇vh
=−X
K∈T
Z
K
(µ|∇unh|p−2∇unh +f′(det∇unh)cof∇unh)· ∇vhdx
− C h
X
E∈E
Z
E
[∇unh ·s][∇vnh ·s]ds, ∀vh ∈Vh,0nc. (4.8)
In the computations, we set wh = utn+1h −unh
n+1−tn and solve X
K∈T
Z
K∇wh· ∇vh
=−X
K∈T
Z
K
(µ|∇unh|p−2∇unh +f′(det∇unh)cof∇unh)· ∇vhdx
− C h
X
E∈E
Z
E
[∇unh ·s][∇vnh ·s]ds, ∀vh ∈Vh,0nc. (4.9) at each time step. Then we compute un+1h = unh + (tn+1 −tn)wh to find the solution of the next step.
Denoting the set of nodes of the triangulation byN, define the norm kwhk by kwhk2 := P
a∈N |wh(a)|2. The numerical algorithm used in our computa- tions is then as follows:
• Step 1. Specify a triangulation Thǫ, the initial values u0h, a constant T OL >0, and set n = 0;
• Step 2. Solve the equation (4.9) to obtain wh;
• Step 3. Compute un+1h =unh + (tn+1−tn)wh and let n=n+ 1;
• Step 4. If kwhk< T OL, stop; otherwise, go to Step 2.6
In Step 3, we choose the time step size δt = tn+1 −tn adaptively in the following manner. For eachn, we computeun+1h using the same value ofδtas in the last step. We accept un+1h ifIh(un+1h )< Ih(unh) and det∇un+1h |K >0 in each element. Otherwise, we decrease the time step by a pre-defined factorα∈(0,1), i.e. we set δt = αδt. This process is done iteratively until un+1h is accepted.
Based in numerical experiments, we work with α = 0.1 and t1 −t0 = 0.02. In most time steps, no reductions of the form δt = αδt are necessary. To avoid the time step becoming smaller and smaller, we reset δt = δt/α every certain number M of steps. In our computations, we set M = 100. With this choice of parameters, the program runs fast in the beginning and finds the minimizers, up to a certain error, after a few thousands of steps.
5. Numerical Experiments and results
In our experiments we consider a two-dimensional body, occupying the unit disc in the plane as its reference configuration, with stored-energy function
W(F) = 2|F|32/3 + (detF−1)2/2 + (detF)−1 (5.1) (it corresponds to f(s) = (s−1)2/2 + 1/s, p = 32 and µ = 1 in (2.3)). We consider radial displacement boundary conditions of the form u(x) = λx for
|x|= 1, with λ varying from 1.1 to 2.5. The initial radius ǫ of the microcavity was given various values from 0.005 to 0.05. The initial deformation specified was u(x)|t=0 =λx.
The above choice of domain geometry and of boundary conditions was made for simplicity and for comparison with previous results. Our method can be applied also to more general geometries and boundary conditions.
6A better stop criteria would bekwhk< T OLkuhk, as suggested by the anonymous referee.
5.1. Validation and efficiency of the algorithm. In this subsection we consider the simple example of a body with a single cavity, located at the centre.
Figure 5.1 shows the initial and deformed configurations when ǫ = 0.01 and λ = 1.8. The original cavity has increased about 100 times its size, which corresponds clearly to a cavitation singularity. Various initial conditions were considered, including some that are not radially symmetric, as those in [29] (e.g.
R0(r) = λr, Θ0(θ) = θ+ 4r(1−r), where (R0,Θ0) and (r, θ) denote the polar coordinates of u0 and x, respectively). The computations using these different initial conditions produced always almost the same deformed configuration.
These results support the long-standing conjecture that for this choice of domain and boundary conditions the minimizer is radially symmetric (see [40, 41]).
Figure 5.2 shows the diminishment of the discrete energy Ihǫ and of the residual kwhkwith respect to the artificial time t. We observe that the energy decreases very fast and almost attains the minimal energy att = 7 (the minimal time step observed in this experiment is 0.002). The residual kwhk decreases exponentially.
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
−1.8 −1.4 −1 −0.6 −0.2 0.2 0.6 1 1.4 1.8
−1.8
−1.4
−1
−0.6
−0.2 0.2 0.6 1 1.4 1.8
Figure 5.1. Reference and deformed configurations forǫ= 0.01, λ= 1.8.
In order to validate and to study the efficiency of our numerical method, we compare our solutions to the two-dimensional minimization problem against the minimizers of the elastic energy among radially symmetric deformations.
Even though the latter are not known explicitly (in the case of compressible elasticity, when W(F) is given by (2.3)), they can be computed numerically with very high precision, making the validation possible. Indeed, by writing
0 5 10 15 14.9
15 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9
t Ihε
0 5 10 15
−5
−4
−3
−2
−1 0 1
t log||wh||
Figure 5.2. The discrete energy Ihǫ and kwhk with respect to the artificial time t.
.
u(x) = u(r)xr, with r=|x|, the original problem (2.9) reduces to
u∈W1,pmin(ǫ,1);u(1)=λ2π Z 1
ǫ
Φ(r, u, u′)rdr, (5.2)
where Φ(r, u, u′) = µp
(u′)2+ ur2p2
+f(uur′), which can be computed using very fine meshes (cf. [21]). Denoting its numerical solution by ˜uh(r), we can define the numerical error for the original problem as
err =
uh−u˜h(r)x r 0,Ωǫ, that is, the L2 distance between uh and ˜uh(r)xr.
Figure 5.3 shows, for various ǫ, the convergence behavior of the compu- tational error with respect to the number Ns of degrees of freedom, which represents the number of sides in the triangulation. For the comparison, we set λ = 1.8 (value at some distance of the critical displacement for cavita- tion, as observed in our computations) and considered the three different cases ǫ = 0.02,0.01,0.005. For each case, solutions were computed using various tri- angulations, with respective values ofNs given by 190,740,2920,11600. For the meshes with the same Ns and different ǫ, the two mesh parameters N and δ, defined in Sections 3.2 and Section 3.3, remained invariant. Figure 5.3 suggests
two conclusions. First, the convergence order of the computational error isNs−1 for all cases. Second, the convergence behavior for the nonconforming finite ele- ment method is not noticeably affected by changes inǫ, confirming our analysis of Section 3.3.
2 2.5 3 3.5 4 4.5
−3
−2.5
−2
−1.5
−1
−0.5
log10Ns log10err
1/1
ε=0.02 ε=0.01 ε=0.005
Figure 5.3. Error vs. degrees of freedom under various ǫ (λ = 1.8).
5.2. Concordance with previous analyses and experiments. In this sub- section we present our results on the dependence of the cavity radius in the de- formed configuration with respect to the displacement parameter λ. We show, in particular, the existence of a limiting profile for this relation as ǫ → 0, as well as the existence of a critical displacement for cavitation, in agreement with the analyses of Ball [4], Sivaloganathan [35], Horgan and Abeyaratne [19] and the experimental and theoretical study of Gent and Lindley [13].
Figure 5.4 shows the relation betweenλ and the cavity radius, for the prob- lem considered in the previous subsection (that of a ball with an ǫ microcavity at the origin, subjected to a radial stretch λ at the outer surface), for different values of ǫ. The cavity radius changes dramatically for values ofλ between 1.4 and 1.5. This corresponds to the critical load for the opening of a cavity in the limit as ǫ→ 0. For further results on the computation of the critical displace- ment and the generation of the approximate bifurcation diagram, we refer to the novel numerical method of Negr´on-Marrero and Sivaloganathan [30].
Figure 5.5 shows the tangential strain in the body for different values of λ.
It verifies that in order to reduce the global energy, a minimizer will experience
